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We study deterministic matrix completion problem, i.e., recovering a low-rankmatrix from a few observed entries where the sampling set is chosen as the edgeset of a Ramanujan graph. We first investigate projected gradient descent (PGD)applied to a Burer-Monteiro least-squares problem and show that it convergeslinearly to the incoherent ground-truth with respect to the condition number\k{appa} of ground-truth under a benign initialization and large samples. Wenext apply the scaled variant of PGD to deal with the ill-conditioned case when\k{appa} is large, and we show the algorithm converges at a linear rateindependent of the condition number \k{appa} under similar conditions. Finally,we provide numerical experiments to corroborate our results.

Nonconvex Deterministic Matrix Completion by Projected Gradient Descent Methods